Charge order breaks time-reversal symmetry in CsV$_3$Sb$_5$

The recently discovered vanadium-based kagome metals $A$V$_{3}$Sb$_{5}$ ($A$~=~K,~Rb,~Cs) exhibit superconductivity at low-temperatures and charge density wave (CDW) order at high-temperatures. A prominent feature of the charge ordered state in this family is that it breaks time-reversal symmetry (TRSB), which is connected to the underlying topological nature of the band structure. In this work, a powerful combination of zero-field and high-field muon-spin rotation/relaxation is used to study the signatures of TRSB of the charge order in CsV$_3$Sb$_5$, as well as its anisotropic character. By tracking the temperature evolution of the in-plane and out-of-plane components of the muon-spin polarization, an enhancement of the internal field width sensed by the muon-spin ensemble was observed below $T_{\rm TRSB}=T_{\rm CDW}\simeq95$~K. Additional increase of the internal field width, accompanied by a change of the local field direction at the muon site from the $ab$-plane to the $c$-axis, was detected below $T^\ast\simeq30$~K. Remarkably, this two-step feature becomes well pronounced when a magnetic field of 8~T is applied along the crystallographic $c-$axis, thus indicating a field-induced enhancement of the electronic response at the CDW transition. These results point to a TRSB in CsV$_3$Sb$_5$ by charge order with an onset of ${\simeq}~95$~K, followed by an enhanced electronic response below ${\simeq}~30$~K. The observed two-step transition is discussed within the framework of different charge-order instabilities, which, in accordance with density functional theory calculations, are nearly degenerate in energy.


I. INTRODUCTION
The emergence of metallic kagome materials featuring an intricate structural lattice and rich diversity of quantum phases has reinvigorated the quest for finding materials with topological phases built from strongly interacting electrons. 1-3 This led to the discovery of a vanadium-based kagome metal family AV 3 Sb 5 (A = K, Rb, Cs), [4][5][6] which was reported to feature a metallic, topological phase at high temperature, anomalous transverse transport properties including the anomalous Hall effect, 7 anomalous Nernst effect, 8 unconventional planar Hall effect, 9 a transition to a highly tunable unconventional superconducting state at low temperatures, [10][11][12][13][14][15] and time-reversal symmetry breaking (TRSB) charge order. The TRSB isotropic 2×2 charge density wave (CDW) order was first suggested by magnetic-field based scanning tunneling microscopy data [16][17][18][19][20][21] and was later widely discussed theoretically. [22][23][24][25]54 Several groups observed a reduced CDW symmetry obtained by a 4×1 charge modulation in the CDW state, 26 which would reduce the rotational symmetry from sixfold C 6 to twofold C 2 .
The combination of zero-field (ZF) and high transverse-field (TF) muon-spin rotation/relaxation (µSR) experiments has provided direct evidence for TRSB below the onset of charge order in KV 3 Sb 5 . 14 Similarly, the appearance of spontaneous fields below the charge ordering temperature was also reported for RbV 3 Sb 5 . 27 ZF-µSR experiments on the sister compound CsV 3 Sb 5 have reported the onset of the TRSB state at T TRSB 70 K, 28 which is lower than the CDW transition temperature T CDW 95 K. A more recent ZF-µSR study of CsV 3 Sb 5 reported the appearance of spontaneous fields below 50 K. 29 In contrast to ZF-µSR experiments, Kerr effect measurements reveal the emergence of a TRSB signal in CsV 3 Sb 5 exactly at T CDW . 30 Consequently, the determination of the true onset of spontaneous fields in CsV 3 Sb 5 as well as their in-plane and the out-of-plane anisotropy are of paramount importance as they should intimately relate to the mechanism of charge order.
In this paper, we utilize the combination of ZF and TF-µSR techniques to probe the µSR relaxation rates in CsV 3 Sb 5 as a function of temperature, field, and angle α between the in-plane component of the muon-spin polarisation and the crystallographic a−axis. The main observation is a two-step increase of the internal field width sensed by the muon-spin ensemble. It consists of a noticeable enhancement at T TRSB 95 K, corresponding to the CDW ordering temperature T CDW , followed by a stronger increase below T * 30 K. An applied magnetic field of 8 T along the crystallographic c−axis further enhances the magnetic response below T CDW , leading to a much more pronounced two-step increase of the internal field width. Furthermore, the local field at the muon site lies within the a-a−plane of the crystal in the temperature range from T CDW to T * . Below T * , the internal field also acquires an out-of-plane component. The absence of the in-plane anisotropy of the internal fields down to 3 K was also detected, while the out-of-plane anisotropy remains strong. Our results provide evidence for time-reversal symmetry-breaking in CsV 3 Sb 5 at the onset of charge order, as well as a non-trivial temperature evolution of the electronic response within the charge ordered state. More generally, these results indicate a strong interplay between magnetic and charge channels in this kagome material.
The paper is organized as follows: Section II describes the sample preparation procedure and details of the µSR experiment. Details of the zero-field and transverse-field µSR data analysis process are given in Section II E. The results of the zero-field and 8 T transversal-field µSR experiments are discussed in Section III. Conclusions follow in Section IV.

A. Sample preparation and characterization
Single crystals of CsV 3 Sb 5 were grown following the procedure described in Ref. 31. Single crystals with dimensions of 3 × 3 × 1 mm 3 were used. As demonstrated in Ref. 31, the CsV 3 Sb 5 single crystals possess an obviously hexagonal shape, which allows one to easily distinguish the main crystallographic axes (a and c).
The X-ray Laue diffraction images of the studied crystals demonstrate the single crystallinity of the material and confirm the correspondence of the main crystal axes to the sample shape. The Laue images for six individual crystals used in "in-plane" rotation experiments are shown in the Supplementary Information. 32 Based on the results of the Laue studies, the crystals were aligned along a and c axes and mounted on a specially constructed µSR sample holder (see Sec. II C and the Supplementary Information, 32 for further details).
The superconducting transition temperature T c was determined by means of ac susceptibility and was found commonly for all crystals to be T c 2.7 K, in agreement with previously published data. 5 The ac susceptibility curves for two sets of crystals used in zero-field and transverse-field µSR experiments are presented in the Supplementary Information. 32

B. Muon spin rotation/relaxation experiments
The muon spin rotation/relaxation (µSR) experiments were carried out at the πM3 and πE3 beam lines by using the GPS, Ref. 34, and HAL-9500 spectrometers (Paul Scherrer Institute, Switzerland). The µSR measurements were performed at temperatures ranging from 3 to 300 K. The 100% spin-polarized "surface" muons with a momentum of p µ 28.6 MeV/c were implanted into the CsV 3 Sb 5 crystals along the c−axis [see Fig. 1  The single-crystalline sample (yellow hexagon) has its c−axis aligned along the incoming muon beam (red arrow). The sample can be rotated within the (x − y) plane by changing the angle α between the in-plane component of the muon-spin polarisation P ab µ and the crystallographic a−axis. In experiments performed on the HAL-9500 spectrometer, the initial muon-spin polarization was aligned perpendicular to the muon-momentum (β 90 o ). The external magnetic field Bext = 8 T was applied parallel to the muonmomentum (Bext pµ) and parallel to the crystallographic c−axis (Bext c). (b) The sample holder for "in-plane" rotation experiments. The angle α can be changed with 10 o step (see text for details).
∼ 5 g/cm 3 , this corresponds to a depth of 0.3 mm. With the CsV 3 Sb 5 crystal thickness of 1 mm, all "surface" muons stop in the sample, so the use of degraders, as it is required in µSR studies of thin single-crystal samples, 33 was not necessary.

C. ZF-µSR experiments
Experiments with zero applied field were performed at the GPS spectrometer. Measurements were made by varying two angles: β, the angle between the initial muon-spin polarization P µ and the muon momentum p µ ;  Fig. 1 (a)] was accessed. The "out-of-plane" experiments were conducted with α and β set to 30 o and 5 o , respectively. The temperature evolution of P c µ component of the muon-spin polarization was measured. Note that β = 5 o corresponds to the smallest possible muon-spin rotated angle at the GPS instrument. 34 In order to vary the angle α, a special sample holder was constructed [ Fig. 1 (b)]. It consists of an aluminum support plate and two sample mounting rings. Each sample ring has 36 holes, which allows for rotation with a 10 o step relative to the support plate. The a− and c−axisaligned CsV 3 Sb 5 crystals were glued on a 25 µm aluminum foil, which was further attached to one of the aluminum sample rings. The second ring was covered by a 25 µm thin layer of Kapton in order to prevent the crystals from falling down inside the cryostat. Note that the Kapton and aluminum foils are fully transparent for the "surface" muons, which allows us to use the advantage of the so-called "Veto" mode. The "Veto" mode rejects the muons missing the sample and, as a consequence, reduces the background of the µSR response to nearly zero (see Ref. 34, for a detailed explanation of the "Veto" mode principle). Photos of six CsV 3 Sb 5 crystals mounted on the muon sample holder and the background estimate are given in the Supplementary Information. 32

D. TF-µSR experiments
Experiments with the magnetic field applied transversal to the initial muon-spin polarisation (TF-µSR experiments) were performed at the HAL-9500 spectrometer. The initial muon-spin polarization was set perpendicular to the muon-momentum [β 90 o , Fig. 1 (a)]. The external magnetic field, B ext = 8 T, was applied parallel to the muon-momentum (B ext p µ ) i.e. parallel to the crystallographic c−axis (B ext c). In these experiments the time evolution of the P ab µ component of the muon-spin polarization [ Fig. 1 (a)] was accessed. Crystals were mounted on the sample-holder made of 99.999% pure silver, and can bee seen in a photo shown in the Supplementary Information. 32

E. µSR data analysis procedure
The zero-field µSR spectra were fitted using the Gaussian Kubo-Toyabe (GKT) relaxation function, 35,36 describing the nuclear moment response, multiplied by an additional exponential term: (1) Here, A ZF 0,i is the initial asymmetry of the i−th positron detector at t = 0, σ GKT is the GKT relaxation rate, and λ is the exponential relaxation rate. Note that Eq. 1 is widely used to analyze the ZF-µSR data in most TRSB µSR studies. 14, [37][38][39][40] The TF-µSR data were analyzed as: Here B int is the internal field at the muon site, φ i is the initial phase of the muon-spin ensemble, γ µ = 2π × 135.5 MHz/T is the muon gyromagnetic ratio, and σ is the Gaussian relaxation rate.
In the above Eqs. 1 and 2, σ GKT and σ relaxation rates mainly account for the nuclear moment contribution, which is assumed to be static within the µSR time window. As discussed previously for KV 3 Sb 5 and RbV 3 Sb 5 , the exponential relaxation rate λ is mostly sensitive to the temperature dependence of the electronic contribution to the muon spin relaxation. 14,27 One cannot exclude, however, subtle effects owing to changes in the electric field gradients in the charge ordered state. 41 The details of the ZF-and TF-µSR data analysis procedure are discussed in the Supplementary Information. 32

III. EXPERIMENTAL RESULTS AND DISCUSSIONS
The parameters obtained from the analysis of ZF-and TF-µSR data are summarized in Figs. 2, 3, and 4. Figure 2 compares the results of the "in-plane" and "out-of-plane" ZF-µSR experiments conducted at α = 30 o . The left and right columns of Fig. 2 represent the parameters obtained from the fits of the P ab µ (t) and P c µ (t) components of the muon-spin polarization, respectively. Figure 3 shows the temperature evolution of the fit parameters obtained from the 8 T TF-µSR experiments. Panels (a), (b), and (c) refer to the temperature dependencies of σ, λ, and the experimental muon Knight shift (B int − B ext )/B ext , respectively.   The experimental results presented in Figs. 2, 3 and 4 are closely related to each other and display distinct features, which will be discussed in the following Sections.
A. Two-step feature and out-of-plane anisotropy of the internal field Figure 2 shows that three of four ZF relaxation rates, namely λ in the P c µ set of experiments (λ c ) and the two Kubo-Toyabe relaxation rates from the P ab µ and P c µ sets of data (σ ab GKT and σ c GKT ) display a sudden change across T CDW 95 K (indicated by the thick grey lines in all four panels of Fig. 2). Namely, both σ GKT suddenly increase, while λ decreases when crossing the CDW transition temperature. The absolute value of the jump, as estimated from the linear fits of the relaxation rate data above and below T CDW , was found to be the same (within the experimental uncertainty) and corresponds to 0.0075 (20)  is associated with the field components from the ac− and/or bc−planes (∆B ⊥ ). Accounting for the experimental data presented in Figs. 2 (b), 2 (d), and 4 (b), this implies that for T * T T CDW the internal field on the muon position has only B components, while for T T * both B and B ⊥ components are present.
The increase of the exponential contribution to the internal field width is also accompanied by a nonmonotonic temperature dependence of the Gaussian contribution to the internal field width. Below T CDW 95 K, the Gaussian relaxation rate changes in two steps.
In the region T * T T CDW , σ ab GKT (T ) decreases, while σ c GKT (T ) stays nearly constant with decreasing temperature. Below T * 30 K, the temperature dependence of both relaxation components change slope and begin to increase.
While the increase of the exponential relaxation λ c (T ) of CsV 3 Sb 5 is consistent with the onset of time-reversal symmetry-breaking at T CDW (T TRSB T CDW ), highfield µSR experiments are essential to confirm this effect, as the ZF and weak TF-µSR data can be subtly affected by the onset of different charge orders even without the presence of TRSB. Following Ref. 14, the application of a high magnetic field leads to a strong enhancement of the electronic contribution to the relaxation rate. Comparison of the relaxation rates obtained in ZF-and TF-µSR experiments confirms that this is actually the case. Indeed, σ ab (T ) measured at B ext = 8 T [ Fig. 3 (a)] reproduces the temperature evolution of σ ab GKT collected in ZF-µSR studies [ Fig. 2 (a)]. The absolute values of both σ ab and σ ab GKT stay almost the same within 10-15%. In contrast, the exponential component is strongly affected by the applied field. There is a factor of ∼ 1.5 − 2 enhancement of λ ab (T ) in the T 30 K region and a full recovery of the λ ab component for temperatures between 30 K T 95 K. This gives rise to a well-pronounced two-step feature at high fields. Note that such a twostep increase of the relaxation rate is also observed in the sister compound RbV 3 Sb 5 . 27 To estimate the onset temperatures associated with this two-step transition, the TF λ ab (T ) was fitted with a double-stage power-law functional form. While this is not a microscopically derived function, it allows us to gain quantitative insight about the temperatures involved. The fit function assumes that each state is characterised by its own transition temperature (T TRSB and T * ), the enhancement of λ(T ) due to spontaneous magnetic fields (∆λ 1 and ∆λ 2 ), and the power-law exponent (n 1 and n 2 ): The solid line in Fig. 3  The combination of ZF-µSR and high-field TF-µSR results on CsV 3 Sb 5 provides an indication of timereversal symmetry-breaking below the onset of charge order T TRSB = T CDW 95 K. This agrees well with the previous reports on KV 3 Sb 5 14 and RbV 3 Sb 5 , 27 indicating that the TRSB effect is strongly connected to the charge-density wave transition for all three members of this kagome metal family. It has to be mentioned, however, that the effects of TRSB in CsV 3 Sb 5 are much less pronounced than for the sister compounds KV 3 Sb 5 14 and RbV 3 Sb 5 27 and might easily be overlooked by less precise measurements. It seems to be especially vital that high quality single crystals are used for the investigation. One way to understand these results is that the internal fields experienced by the muons are generated by orbital currents associated with a complex CDW order parameter. [22][23][24]47,48 Within this framework, muons can couple to the fields generated by these loop currents, resulting in an enhanced internal field width sensed by the muon-spin ensemble. A direct connection between the orbital current patterns and the observed internal fields remains a challenge. The first attempt in calculating possible field directions was made in Ref. 28 by considering a few possible orbital current configurations which are allowed by symmetry. 49 At the moment, it is difficult to proceed deeper into the subject. Further theoretical studies, including the exact determination of the muonstopping site(s) and possible configurations of the orbital currents are needed.
The increase of the exponential relaxation rates below the characteristic temperature T * 30 K is suggestive of another transition that modifies the loop currents formed at T TRSB = T CDW 95 K. In Ref. 28 the low-temperature increase was interpreted as a change in symmetry of the orbital currents within the same chiral flux state. However, there are experimental indications that some kagome metals (including CsV 3 Sb 5 ) may exhibit two charge-order transitions. 50,51 For instance, the coexistence of the tri-hexagonal and Star-of-David CDW patterns in CsV 3 Sb 5 was reported by ARPES. 51 Similarly, NMR/NQR experiments point to the Star-of-David CDW at high temperatures, followed by an additional charge modulation below ∼ 40 K. 50 Breaking of the sixfold symmetry of the CDW state was reported experimentally for CsV 3 Sb 5 . 19,52,53 More broadly, the pressuredependent µSR data on the RbV 3 Sb 5 compound also indicates two CDW transitions. 27 Our previous low-temperature, pressure-dependent µSR data on CsV 3 Sb 5 revealed a strong change in the superfluid density within the CDW phase as pressure was varied. 46 Combined with the first-principles calculations, this was interpreted as indicative of a change in the zerotemperature CDW ground state as a function of pressure. Such a behavior might also be consistent with a change in the CDW state as a function of temperature for zero applied pressure. Table I reports our density functional theory (DFT) results from Ref. 46 for the relative energies of three of the possible CDW states at ambient pressure -namely, the planar tri-hexagonal, staggered tri-hexagonal, and superimposed tri-hexagonal Star-of-David (see Ref. 55 for the schematics of each CDW configuration). As follows from Table I, the energy differences between these states are so small (about 5 meV) that the issue of which CDW state has the lowest energy is likely to be affected by finite-temperature effects. 56,57 Such effects might be associated with either the "electronic temperature" within DFT or the entropy contribution to the free-energy, which is not captured by DFT. 55 Moreover, phonon modes associated with other CDW states with additional modulation along the z−axis not considered here (e.g. a 2×2×4 configuration) are also unstable, expanding the landscape of possible CDW configurations even further. 58 It is important to note, however, that these non-spinpolarized first-principles calculations that do not take spin-orbit coupling into account refer only to the real component of the complex CDW order parameter. As such, they do not capture the role of orbital currents. Furthermore, our µSR results show that not only timereversal symmetry is broken below T CDW , but that the internal magnetic fields rotate out of the plane below T * . Therefore, a full understanding of the two-step transition observed here will require a more in-depth analysis of the role of the imaginary component of the complex CDW order parameter. While first-principles calculations of orbital currents are challenging and likely cannot be captured at the generalized gradient approximation level, phenomenological approaches reveal an interesting connection between the real and imaginary components, 23,24 which deserve further exploration.   Fig. 1). The temperature dependencies of both relaxation rates (σ GKT and λ) do not depend on the angle α within experimental accuracy. Thus, the internal field width seems to be isotropic within the kagome plane, while it acquires a strong out-of-plane anisotropy. At the present stage it is difficult to establish a direct relation between the in-plane isotropic internal fields and the symmetry of the CDW. Further theoretical and experimental studies, including the exact determination of the muon-stopping site(s), are required.

IV. CONCLUSION
In conclusion, a combination of zero-field and high transverse-field muon-spin rotation/relaxation experi-ments were performed on the CsV 3 Sb 5 representative of the kagome superconducting family AV 3 Sb 5 (A =K, Cs, Rb). The in-plane and out-of-plane electronic responses as a function of temperature and magnetic field in the normal state were studied. An enhancement of the width of the internal magnetic field distribution sensed by the muon-spin ensemble was found to coincide with the onset of the charge ordering transition, thus suggesting that the CDW order breaks time-reversal symmetry. A magnetic field of 8 T applied along the crystallographic c−axis further promotes the electronic response below T CDW , leading to a more clearly pronounced twostep increase of the internal field width at the characteristic onset temperatures T TRSB = T CDW 95 K and T * 30 K, respectively. The local fields at the muon stopping site, which are potentially created by loop currents, were found to be confined within the crystallographic ab-plane for temperatures between T CDW and T * , while they possesses a pronounced out-of-plane component below T * . Rotation of the crystals around the c−axis suggests that the internal field remains isotropic within the kagome plane, in sharp contrast to the highly anisotropic out-of-plane behaviour. Our results indicate a rich electronic response promoted by complex charge order realized in the kagome superconductor CsV 3 Sb 5 and provide useful insights into the nature of the timereversal symmetry-breaking charge density wave order.

Acknowledgments
The work was performed at the Swiss Muon Source (SµS), Paul Scherrer Institute (PSI, Switzerland). The work of R.G. was supported by the Swiss National Science Foundation (SNF Grant No. 200021-175935    The Laue x-ray images of six CsV 3 Sb 5 crystals from Fig. 1 are presented in Fig. 2. All images clearly demonstrate the hexagonal in-plane crystal structure of CsV 3 Sb 5 . The single crystallinity of the material and the correspondence of the main crystal axes to the sample shape are visible.

FIG. 2:
The Laue x-ray images of the six CsV3Sb5 singlecrystals presented in Fig. 1. The images demonstrate the single crystallinity of the material and correspondence of the main crystal axes to the sample shape.

II. AC SUSCEPTIBILITY RESPONSE OF CsV3Sb5 SINGLE-CRYSTALS
The superconducting response of CsV 3 Sb 5 crystals was studied via ac susceptibility (ACS) experiments. CsV 3 Sb 5 single crystals were placed inside the opened pressure cell (in our ACS experiments the pressure cell was not sealed and it was simply used a sample container). The ACS setup and the pressure cell are described in Refs. 1,2.

IV. THE BACKGROUND ESTIMATE
In µSR experiments, part of the muons may not hit the sample and stop in the sample holder, cryostat windows, cryostat walls, etc. These muons contribute in the background response, which must be known prior to performing the data analysis.

A. The background contribution in GPS experiments
In experiments performed at the GPS spectrometer, the samples (CsV 3 Sb 5 single-crystals) were mounted on the holder shown in Fig. 4. The background was estimated from measurements made in the superconducting state. The external magnetic field B ext = 10 mT was applied parallel to the crystallographic c−axis at T = 5 K, i.e. above the superconducting transition temperature T c 2.6 K [see Fig. 3 (a)]. The angles α and β i.e. the angle between the crystallographic c−axis and the in-plane component of the muon-spin polarization P ab µ and the angle between the muon momentum p µ and the initial muon-spin polarization P µ , we kept at α = 30 o and β = 45 o , respectively.
The Fourier transforms of TF-µSR time spectra, representing the internal field distribution P (B), are shown in Fig. 6. The panel (a) corresponds to P (B) obtained (1) Here A 0,S and A 0,BG are the initial asymmetries, while P S (t) and P BG (t) are the time evolution of the muon-spin polarizations of the sample and the background component, respectively. The background contribution was approximated by a cosine decay function: Here γ µ = 2π135.5 MHz/T is the muon gyromagnetic ratio, σ BG is the Gaussian relaxation rate and φ is the initial phase of the muon-spin ensemble. In order to account for the asymmetric field distribution P (B) caused by formation of the flux-line lattice (FLL) in the superconducting state, the sample contribution was described using the Skewed Gaussian (SKG) distribution function: 3 Here B 0 is the field corresponding to the maximum of P (B) distribution of SKG(t) function. 3 The fit of Eq. 1 with the sample and the background components described by Eqs.  two-component signal was observed within the full range of temperatures. Figure 7 shows the field distribution P (B) measured at T = 20 K. The solid line corresponds to the two component fit by means of Eq. 1. The background and the sample component were described by the aforementioned Eq. 2 and by Eq. 2 from the main text, respectively. The background contribution is visualized as a difference between the two-component fit and the sample contribution (see the pink area at the inset to Fig. 7). The width and the position of the background signal were found to be temperature independent, as expected, and they were kept fixed during the analysis (A 0,BG /A 0 0.2, σ BG 0.16 µs −1 ).

V. THE PRESENCE OF NUCLEAR AND ELECTRONIC COMPONENTS IN µSR RESPONSE
Validity of the data analysis procedure considering the presence of both -the electronic and the nuclear components -was checked for the set of ZF-µSR data collected at T = 5 K for two different components of the muonspin polarization [P ab (t) and P c (t)]. The red and black curves correspond to the data collected at positron detec-tors staying in-phase (0 o ) and out-of-phase (180 o ) to the corresponding initial component of the muon-spin polarization. Three different fit types of Eq. 1 from the main text to the experimental data were performed: 1. Both, λ and σ GKT , remain free, i.e., stay fitted [panels (a) and (d)] 2. σ GKT stays free and λ is fixed to '0' [panels (b) and (e)]. 3. λ remains free and σ GKT = 0 [panels (c) and (f)]. The results of the fit of Eq. 1 from the main text to the data are presented in Fig. 8 by solid lines. The results of the fit of Eq. 1 from the main text to the muon data recorded at T = 5 K for two different components of the muon-spin polarization P ab (t) and P c (t). The fitted relaxation rates are summarised in Table I.
The goodness of each fit type was evaluated by calculating the values of the normalized χ 2 norm (the sum of the mean squared deviations divided by the number of degrees of freedom). Note that, in their normalized form, χ 2 norm takes into account the decreased number of the fit parameters for cases when one of the relaxation rates (λ or σ GKT ) was fixed to zero.
The results presented in Fig. 8 and Table I imply that the experimental data require the presence of both -the exponential λ and the Gaussian Kubo-Toyabe σ GKTrelaxation components. Exclusion of either λ or σ GKT leads to substantial increase of χ 2 norm . This agrees, therefore, with the results of Ref. 4, where the ZF-µSR data of KV 3 Sb 5 representative of AV 3 Sb 5 family were described by using both -σ GKT and λ relaxation components. A similar approach was used in the majority of other TRSB µSR studies as reported e.g.
in Refs. 5-14. Our results would also suggest that the analysis of CsV 3 Sb 5 and KV 3 Sb 5 ZF-µSR data from Refs. 15 and 16, where only the Gaussian Kubo-Toyabe term was taken into account, need to be reconsidered. Both static and fluctuating internal magnetic fields can have an effect on the exponential muon-spin polarization relaxation rate as is measured in ZF experiments. In order to differentiate between static and fluctuating internal magnetic fields, the longitudinal-field (LF) experiments with the externally applied magnetic field B ext = 3 mT applied along the P ab µ component of the muon-spin polarization were conducted. The results reveal that the Gaussian Kubo-Toyabe component observed in ZF-µSR is absent, as expected when the muonspins are decoupled from the local field, while the exponential relaxation rate λ remains at a certain non-zero value. Figure 9 compares the exponential relaxation rates as they observed in ZF and LF set of µSR experiments. From the data presented in Fig. 9 two important points emerge: (i) The LF relaxation points stay systematically lower the ZF ones. This suggests that 3 mT magnetic field leads to partial, but not complete, decoupling of muon-spins from internal fields. (ii) The LF relaxation rate decreases nearly linearly within the full temperature range studied, i.e. from 3 up to 300 K. This imply that the dynamical component on the exponential relaxation is indeed present. However it is not related to the appearance of the charge density order below T CDW 95 K, as well as to an additional enhancement of the ZF relaxation below T * 30 K.